The Logic Algebra On The Filter And Reverse Triple I Method Unified Form

In order to set a rigorous logic foundation for fuzzy reasoning,many kinds of logical algebras have been developed rapidly.MTL-algebra which serves as the corresponding semantics of the logical system MTL was proposed by Esteva and Godo in 2001.We can obtain IMTL-algebra,which is equivalent to BR₀-algrbra,by adding to MTL-algebra the involutive law,while NM(?)-algebra can be gained by adding two algebraic equalities into IMTL-algebra.Go a step further,MV-algebra and R₀-algrbra are two famous extensions of NM(?)-algebra,the former is the algebra matching with(?)ukasiewicz system while the latter is the algebra corresponding to L^* system proposed by professor Wang Guojun.So BR₀-algrbra and NM(?)-algebra have been paid a great deal of attention because both of them form a connecting link between what comes before and what goes after.Since completeness is a vital property of a formal system,which reflects the harmony between the syntax and the semantics and filters of logical algebras play an important role in the proof of the completeness of the corresponding logical system,therefore it is necessary to investigate and discuss properties of filters in logical algebras.As the theoretical foundation of fuzzy control,fuzzy reasoning has two foundmental models,i.e.,FMP and FMT.With regard to FMP problem,Zadeh put forward a method named CRI in 1973 to solve it.In 1999,Professor Wang proposed the fully implicational tripleⅠmethod to make up defects and inadequacies of CRI method,and the unified forms of tripleⅠmethod is obtained based on the regular implication operators,which makes it possible to establish an unified framework for approximate reasoning in diverse logical systems.In 2002,Professor Song took as a starting point in consideration the minimum of elements in the base of fuzzy rule when designing fuzzy system under a given precision and posed the inverse tripleⅠmethod. Based on R₀ implicational operator and(?)ukasiewicz implicational operator,Song and Qin gave the solution expressions of FMP and FMT problems respectively.A natural question then arises:Can we give a unified solution expression of inverse tripleⅠmethod with respect to R₀ implicational operator and(?)ukasiewicz implicational operator? This paper will give a positive answer to the questions above.This paper is composed of four parts:Chapter 1 is an introduction which summarizes the emergence,development of the theory of logical algebras and the function of filters in logical algebras and the emergence,development of tripleⅠmethod and inverse tripleⅠmethod of fuzzy reasoning.Chapter 2 first introduces the concepts of a normal MP-filter and a Boolean MP-filter and some properties are discussed.Secondly,the necessary and sufficient conditions under which a filter is a normal MP-filter or a Boolean MP-filter are given.Lastly,the equivalent characterization of a normal BR₀-algebra is obtained.Chapter 3 first introduces the concepts of MP-filters and prime filters. Secondly,the foundmental properties of filters and prime filters in NM(?) algebras are studied and the equivalent characterizations of prime filter are given.Lastly,we enrich a topological structure on the set of all prime filters.Based on the R₀ implicational operator and the(?)ukasiewicz implicational operator,the fourth part of the paper investigates the inverse tripleⅠmethod and the inverseα-tripleⅠmethod of fuzzy reasoning,a unified solution expression for R₀ implicational operator and(?)ukasiewicz implicational operator are obtained.